SARIMAX Time Series Models with R Code

Here’s a comprehensive guide to SARIMAX in R with examples and interpretation:

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1. What is SARIMAX?

SARIMAX = Seasonal AutoRegressive Integrated Moving Average with eXogenous variables

• SARIMA: Handles seasonality and trends
• X: Includes external predictors (covariates)

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2. Basic Syntax in R

library(forecast)

# Fit SARIMAX model
model <- Arima(y,
               order = c(p, d, q),           # non-seasonal ARIMA order
               seasonal = c(P, D, Q, S),     # seasonal order (S = period)
               xreg = xreg_data)             # exogenous variables

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3. Example 1: Simple SARIMAX with One External Variable

library(forecast)
library(ggplot2)

# Create sample data
set.seed(123)
n <- 120
time <- 1:n

# Main time series with trend + seasonality
y <- 50 + 0.3*time + 10*sin(2*pi*time/12) + rnorm(n, 0, 5)

# External variable (e.g., marketing spend)
x1 <- 20 + 0.1*time + rnorm(n, 0, 3)

# Convert to time series object
y_ts <- ts(y, frequency = 12)
x1_ts <- ts(x1, frequency = 12)

# Fit SARIMAX model
sarimax_model <- Arima(y_ts,
                       order = c(1, 1, 0),      # Remove MA term
                       seasonal = c(1, 1, 0),   # Remove seasonal MA
                       xreg = x1_ts)

summary(sarimax_model)

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4. Example 2: Multiple External Variables

# Add second external variable
x2 <- 15 + 0.05*time + rnorm(n, 0, 2)
xreg_matrix <- cbind(x1_ts, x2_ts)

# Fit model with multiple external variables
sarimax_model2 <- Arima(y_ts,
                       order = c(1, 1, 1),
                       seasonal = c(1, 1, 1),
                       xreg = xreg_matrix)

summary(sarimax_model2)

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5. Model Diagnostics

# Residual diagnostics
checkresiduals(sarimax_model)

# Coefficients and significance
coef(sarimax_model)
sqrt(diag(sarimax_model$var.coef))  # standard errors

# Confidence intervals
confint(sarimax_model)

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6. Forecasting with External Variables

# Create future values of external variables
future_periods <- 12
x1_future <- ts(30 + 0.1*(121:132), frequency = 12)
x2_future <- ts(20 + 0.05*(121:132), frequency = 12)
xreg_future <- cbind(x1_future, x2_future)

# Generate forecasts
forecast_result <- forecast(sarimax_model2,
                           h = future_periods,
                           xreg = xreg_future)

# Plot results
autoplot(forecast_result) +
  ggtitle("SARIMAX Forecast with External Variables")

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7. Automated Model Selection

# Let auto.arima select best SARIMAX model
auto_model <- auto.arima(y_ts,
                        seasonal = TRUE,
                        stepwise = TRUE,
                        approximation = FALSE,
                        xreg = xreg_matrix)

summary(auto_model)

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8. Real Dataset Example (AirPassengers with exogenous var)

# Using AirPassengers dataset
data("AirPassengers")

# Create dummy external variable (e.g., economic index)
set.seed(123)
economic_index <- 100 + 0.5*time(AirPassengers) + rnorm(length(AirPassengers), 0, 10)

# Fit SARIMAX
air_model <- Arima(AirPassengers,
                  order = c(0, 1, 1),
                  seasonal = c(0, 1, 1),
                  xreg = economic_index)

# Forecast with assumed future economic index
future_econ <- 150 + 0.5*(1961 + (0:11)/12)  # 1961 values
air_forecast <- forecast(air_model,
                        h = 12,
                        xreg = future_econ)

autoplot(air_forecast)

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9. Interpretation of Coefficients

For the model:

# If output shows:
# ar1 = 0.85, ma1 = -0.32, sar1 = 0.72, sma1 = -0.45, xreg1 = 0.65

Interpretation: - ar1 = 0.85: Strong positive autocorrelation (persistence) - ma1 = -0.32: Negative momentum effects - sar1 = 0.72: Strong seasonal autocorrelation - sma1 = -0.45: Negative seasonal momentum - xreg1 = 0.65: 1-unit increase in external variable increases y by 0.65 units

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10. Important Notes

• Stationarity: Ensure both y and xreg variables are stationary (use differencing if needed)
• Correlation ≠ Causation: External variables should theoretically make sense
• Model Validation: Always check residuals for autocorrelation
• Overfitting: Avoid too many external variables relative to data length

This gives you a solid foundation for implementing SARIMAX models in R!